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Mathematics Teacher Education and Development 2018, Vol 20.2,
Published online July 2018 Author draft MERGA
Middle school mathematics teacher preparation in a
Chinese and an Australian university: Different starting
mathematics knowledge
Stephen Norton
Qinqiong Zhang
Griffith University
Wenzhou University
Received: 4 January 2017 Accepted: 18 January 2018
© Mathematics Education Research Group of Australasia, Inc.
There is increasing concern with respect to the quality of teacher education preparation processes in the
West. This study used case study methods to examine learning opportunities and knowledge of
mathematics in a sample of 192 Australian trainee middle school teachers and 94 Chinese Bachelor of
Science trainee teachers. It was found that the Chinese teacher preparation program ensured the
prospective teachers had mastery of basic facts and processes and extended opportunities to deepen
this knowledge and connect this mathematics knowledge to pedagogy. Many of the Australian trainee
teachers had struggled with the same material and had limited opportunity to remediate this situation
prior to commencing classroom engagement. The implications are discussed with regard to program
structure and academic governance within the study institutions.
Keywords: initial teacher education . mathematics knowledge . middle school
Introduction
It has been convincingly argued that quality education at all levels contributes to a nation’s
economic prosperity and social capital (e.g., Dinham, 2015; Keeling & Hersh, 2012; Marginson,
2002, 2006; Sahlberg, 2006, 2007; Sahlberg & Oldroyd, 2010). That is, the investment in human
capital via increasing knowledge increases labour productivity as well as social well-being and
stability. Questions are starting to be asked as to whether the political and institutional
frameworks effectively promote this priority in the English-speaking West (e.g., Australia,
Canada, England, New Zealand, USA). According to a growing literature base there is need for
reform of tertiary institutions in general and education in particular (e.g., Chang, 2002; Keeling &
Hersh, 2012; Kotzee, 2012; Marginson, 2002, 2006; Meyers, 2012; Sahlberg, 2006) since there is a
perception of a drift away from expertise and fluency with professional knowledge associated
with disciplines.
With respect to teaching and learning mathematics it has long been recognised that deep
knowledge of mathematics is necessary for effective teaching (e.g., Australian Academy of
Science, 2015; Australian Institute for Teaching and School Leadership [AITSL], 2012; Ball, Hill,
& Bass, 2005; Burghes & Geach, 2011; Cai, Mok, Reddy, & Stacey, 2016; Goulding, Rowland, &
Barber, 2002; Hattie, 2009; Hill, Rowan, & Ball, 2005; Krainer, Hsieh, Peck, & Tatto, 2015; Masters,
2009; Teacher Education Ministerial Advisory Group [TEMAG], 2014; U.S. Department of
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Education, 2008). This study examines the processes of middle school mathematics teacher
preparation in two institutions, one in Wenzhou China and one in Brisbane Australia, in order to
gain insights from two different teacher education systems. It is not intended to claim that the
institutions are necessarily representative of each nation. It is up to the reader to consider the
transferability of the data and findings to their circumstances. The review of literature and course
and program structures provided in the results section assists in this endeavour.
The Importance of Content Knowledge for Teaching Mathematics
The relationship between knowing mathematics and teaching mathematics has a long history and
nuances underpinning the theoretical models have been refined. Shulman (1986) used the term
mathematical content knowledge (MCK) which included mathematical concepts, fundamental
assumptions, definitions, and procedures. Shulman distinguished MCK from pedagogical
content knowledge (PCK). PCK included curriculum knowledge, knowledge of students,
representations of content, analogies, models, and explanations of mathematics that make it
understandable. Some authors have questioned whether or not it is possible to make a clear
distinction between subject knowledge and PCK since all mathematical knowledge has
pedagogical underpinnings (McEwan & Bull, 1991; Stones, 1992). Lannin et al. (2013) described
PCK as “the most useful forms of representations, analogies… but as subject-specific knowledge
of curriculum, learners, assessment, and instructional strategies…” (p. 423). The link between
depth of relevant mathematical knowledge and teaching effectiveness has been well documented.
Ball et al. (2005) commented: “That the quality of mathematics teaching depends on teachers’
knowledge of content should not be a surprise” (p. 14). These authors went on to claim of United
States teachers that “the mathematical knowledge of teachers is dismayingly thin… we are failing
to reach reasonable standards with most of our students, and most of those students become the
next generation of adults, some of them teachers” (p. 14). The Teacher Education and
Development Study in Mathematics (TEDS-M) noted that “knowledge of content to be taught is
a crucial factor in influencing the quality of teaching” (Tatto et al., 2008, p. 19). In the UK, Burghes
and Geach (2011) reported that “a prerequisite to be an effective teacher of mathematics, is that
you are confident and competent in mathematics at a level significantly above that which you are
teaching” (p. 17). More recently, Gess-Newsome (2013) suggested that “Teachers’ content
knowledge for teaching mathematics (CKT-M) significantly and positively predicted student
achievement.… The only variable that approached CKT-M in explaining student achievement
was students’ socioeconomic status” (p. 258). The debate is not so much that teachers should have
a deep understanding of mathematics, but just what level of mathematics is necessary for
particular levels of teaching. Ball (1990), Hill et al. (2005), and more recently Speer, King, and
Howell (2015) have noted mathematical majors or the equivalent do not necessarily bestow
prospective teachers with depth of understanding of the concepts they will teach. The mastery of
rules and procedures is not enough: deep understanding of the structures underpinning the
mathematics to be taught is needed. As Ball et al. (2005) commented, there is a need for “teachers
to have a specialised fluency with mathematical language, with what counts as a mathematical
explanation, and with how to use symbols with care” (p. 21). Ball et al. (2005) stated that
specialised fluency with mathematical language is a starting point for being able to construct
mathematical explanations. The specialised language of middle years’ mathematics is related to
order convention, index convention, whole-number place-value structures, the detail of algebraic
processes, and conventions associated with logarithms. The need for depth of knowledge in order
to provide necessary scaffolding has been supported by other studies (e.g., Englemann, 2007;
Kirschner, Sweller, & Clark, 2006; Owen & Sweller, 1989). Needless to say, it is difficult to be
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effective in enacting explicit instruction if the teacher has a limited understanding of the material,
something of which the father of social-cultural principles, Vygotsky, was well aware.
While the exact relationship between depth of content knowledge and pedagogical content
knowledge remains an area of development, the emerging literature indicates that a strong
subject-specific knowledge supports the development of PCK (Krauss et al., 2008). These authors
used pen-and-paper mathematics tasks to assess MCK and probe PCK. Their test items were
samples of the concepts they were expected to teach. For example: “How does the surface area of
a square change when the side length is tripled? Show your reasoning” (MCK) and “Note down
different ways of solving this problem” (PCK); “Is it true that 0.999999…=1?” (MCK) and “Please
give detailed reasons for your answers” (PCK) (p. 720). With respect to the use of pencil-and-
paper testing of teachers and prospective teachers, Ball et al. (2005) defended “testing teachers,
studying teaching or teacher learning, at scale, using standardised student achievement
measures” (p. 45).
Support for teachers to have fluency with MCK, including fluency with basic facts and
procedures as the foundation of broader mathematical success, comes from educational theorists
who claim that mathematics is a hierarchical body of knowledge built upon specialised facts,
procedures, and language (Bernstein, 1999, 2000; Muller, 2000, 2009; Muller & Taylor, 1995).
Bernstein used the term “esoteric discourse” while Muller and Taylor (1995) described the nature
of mathematics as “sacred and profound” (p. 263). The hierarchical nature of mathematics means
that students who are not fluent in whole-number numeration and computation will find it
incredibly difficult to succeed with fraction numeration and computation and algebra
computation. Similarly, students who are not fluent in linear algebra conventions will find
calculus a mystery. Cognitive load theorists support this position. The example Sweller (2016)
gives is to “solve for ‘a’ in (a+b)/c = d”. This is a relatively trivial Year 9 standard question where
the student could apply reverse order of operations by first multiplying both sides of the equal
sign by c and then subtracting b from both sides. The point is that, if the student does not know
order of operation convention, solving for “a” is very difficult, in fact near impossible. Recently,
Hattie and Donoghue (2016) used a similar argument and described the retention of accurate
detail (surface learning) or “lower level learning” as a necessary foundation to higher level
problem solving and creative thought. This positon is supported by the Australian Academy of
Science (2015), the Organization for Economic Co-operation and Development (OECD) (2014),
and Klein (2005). Every modern mathematics curriculum, including the current Australian
curriculum, is structured and sequenced in such a way that acknowledges the essentially vertical
nature of mathematics. The Australian Curriculum: Mathematics (Australian Curriculum,
Assessment and Reporting Authority [ACARA], 2012) has a proficiency strand “Fluency” (p. 5).
Fluency is defined thus:
Students develop skills in choosing appropriate procedures, carrying out procedures flexibly,
accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily.
Students are fluent when they calculate answers efficiently, when they recognise robust ways of
answering questions, when they choose appropriate methods and approximations, when they
recall definitions and regularly use facts, and when they can manipulate expressions and equations
to find solutions. (ACARA, p. 6)
Cognitive load theory helps us gain insight as to why a teacher’s fluency and depth of content
knowledge has a positive impact on measures of PCK and student learning. Sweller (2016)
classifies knowledge into two forms: biological primary and secondary. Sweller contends that
primary knowledge includes learning to speak and listen, recognise faces, and engage in generic
cognitive processes such as problem solving by using solution knowledge of related problems.
Such biologically primary knowledge tends to be acquired “without explicit tuition from others”
(Sweller, 2016, p. 360). In contrast, almost all of what is learned in educational institutions is
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classified as secondary knowledge. The acquisition of secondary knowledge is conscious,
relatively difficult, and effortful (Kirschner, Verschaffel, Star, & Van Dooren, 2017). Sweller points
out that attaining secondary knowledge is greatly assisted by explicit instruction; he uses
cognitive load theory and the well-known capacity and duration of working memory to explain
the importance of quality explicit instruction in the learning process. Chen, Kalyuga, and Sweller
(2016, p. 28) reported that, “for the learning of complex materials, explicit instruction is essential
for novice learners”. A further insight is that secondary knowledge is domain specific (Tricot &
Sweller, 2014) and includes conceptual and procedural information. Learning mathematics is
very domain specific (Kirschner et al., 2017). It has long been recognised that deliberate practice
is necessary for the acquisition of expert knowledge (Ericsson, Krampe, & Tesch-Romer, 1993).
Cognitive load theory offers an explanation for the call for mathematics teachers to have a
relatively high MCK, a position that is broadly supported in the mathematics teacher education
literature cited above. Teaching is a very complex and fluid working environment with very high
secondary as well as primary knowledge demands. In a complex classroom environment a
teacher who is taxing her or his short-term memory capacity to think through mathematical
solutions is likely to be unable to take account of a myriad of other critical classroom
responsibilities including classroom management or taking account of individual students’
learning needs. One of the claimed outcomes of low levels of MCK is a lack of significant
deviation from the pedagogy that they experienced during their own school days (Ball, 1988).
That is, teachers with low MCK will tend to be less willing to take risks and explore new ways of
scaffolding mathematics. The prevalence of shallow teaching has manifested itself in “the school
mathematics tradition” (Gregg, 1995). Gregg describes this as a pattern of teaching that
emphasises rote, procedures, and shallow explanations that do not readily assist students to
develop deep conceptualization. Gregg reported that this tradition was “ubiquitous and robust”
(p. 444). Further, to attempt to teach mathematics without personal fluency and depth of
knowledge is likely to be stressful and to impact negatively on self-efficacy (Henson, 2001; Watt
& Richardson, 2013) and, in this way, negatively impact on student learning. Put simply, taking
equality of primary knowledge such as the ability to communicate to students, an expert in their
discipline is likely to outperform a novice in their domain irrespective of differences in their
working memories (Sweller, 2016). A teacher who is not fluent in basic facts, procedures, and
language cannot be considered an expert.
There are different ways to ensure that teachers are well prepared to teach, and all of the top
nations have systems that insist on depth of discipline knowledge, which includes fluency with
facts and processes. In China, South Korea, Japan, and Singapore, the use of high-stakes testing
is a part of the checks and balances to ensure quality (Burghes & Geach, 2011; Tatto, Rodriguez,
& Lu, 2015). With respect to Chinese education, Wang, Cai, and Hwang (2004) reported “a high
degree of instructional coherence as a distinguishing feature in Chinese classrooms… and other
East Asian countries such as Japan” (p. 112). Finland has achieved very substantial improvements
in children’s mathematical outcomes as measured on international tests without the use of high-
stakes testing of children or their teachers. Rather, a more holistic approach to education is
enacted there with a cohesive and centrally organised emphasis on high standards of knowledge
at all levels and a trust in individual teachers’ professionalism (Sahlberg, 2011a). In East Asian
and Finnish systems, coherence is founded on sustained and in-depth teacher preparation where
specific discipline knowledge is central (Burghes & Geach, 2011; Fan, Miao, & Mok, 2004;
Sahlberg, 2011a; Tatto et al., 2015). These authors also noted that this preparation included close
collaboration between schools, tertiary providers, government financing, and bodies accrediting
teacher training programs. Common descriptors of high-functioning educational systems
included that they had selective entry requirements, cohesive structures, and stringent
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graduation requirements. Since the two case study sites in this study are located in China and
Australia, it is worth expanding on the literature related to mathematics learning therein.
Teacher Preparation in China and East Asia
Over the past two decades, teacher training in China has been the subject of strong government
intervention to ensure that teachers are suitably qualified (Burghes & Geach, 2011). There is
considerable competition for teaching positions in China as teaching is seen as a respected
occupation, and this leads to competition that enhances standards. Ma (1999) and Li, Zhao,
Huang, and Ma (2008) suggest that strong basic content knowledge (MCK) has been the
foundation of quality mathematics teaching in China in recent decades. Dai and Cheung (2004)
described the wisdom of traditional mathematics teaching as based upon “concise explanation of
mathematical concepts with abundant practice” (p. 3). Similarly, Fan et al. (2004) reported that
Chinese teachers emphasised fluency and proficiency with basic mathematics as a prelude to
deeper problem solving. Consistent with Ball et al. (2005) and Hill et al.’s, (2005)
recommendations, understanding the subject matter is a first step in developing the specialised
mathematics knowledge and skills used in teaching. The epistemology reflected by Chinese
teacher education processes is consistent with cognitive load theorists (e.g., Chen et al., 2016;
Sweller, 2016).The literature on Chinese mathematics teacher preparation indicates that the
Chinese education processes start with the development of advanced mathematical knowledge
(MCK) (e.g., Gu, Huang, & Marton, 2004; Fan et al., 2004; Lai & Murray, 2012; Li, 2004; Li et al.,
2008; Zhang, Li, & Tang, 2004) and build upon this for the development of pedagogy (PCK).
Further, these authors concluded that China has established a unified pre-service teacher
education system with considerable consistency across the country as well as systematic ongoing
professional development programs for in-service teachers. Similar findings have been reported
in other East Asian studies that include nations such as Korea, Japan, Singapore, and Taipei (e.g.,
Tatto et al., 2015).
Teacher Preparation in Australia and the West
From the Western perspective, a matter of concern for some considerable time has been the depth,
or lack thereof, of MCK of some teachers (e.g., Ball, 1988, 1990; Ball et al., 2005) and much of this
concern has been focused on primary teachers (e.g., Brown, McNamara, Hanley, & Jones, 1999;
Burghes & Geach, 2011; Wragg, Bennett, & Carre, 1989). Those studies that have undertaken a
review of Western middle school trainee teachers’ mathematics knowledge had similar concerns
(e.g., Burghes & Geach, 2011; Hine, 2015; Krainer et al., 2015; Ma, 1999). The Krainer et al. data
are particularly concerning for U.S. middle school teachers. In Australia, Masters (2016)
commented that a critical factor with respect to teacher preparation is that entry vetting is very
lax, especially with regard to primary teaching. Part of the reason for this is that teaching in
general in Australia is not seen as a high-status profession and thus does not attract top graduates.
More specifically, with respect to secondary mathematics teaching within the state of
Queensland, The Queensland Audit Office (QAO) (2013) reported that 49% of teachers teaching
classes of mathematics were out of field. That is, they had no formal training in mathematics
teaching. One of the implications of this finding is that almost all trainee teachers who qualify to
teach mathematics are in a very good positon to acquire employment. This factor would seem to
militate against the free market forces driving reform through skilled labour competition for
employment.
Henderson and Rodrigues (2008), Hine (2015), and Kotzee (2012) reported that it has become
a tradition in teacher education courses at Western universities to focus on big-picture curriculum
issues including thinking skills, problem solving, and team work, and to largely assume that
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students have a basic knowledge of content. Poulson (2001) provides a rationale for not being too
concerned with a teacher’s capability to articulate the detail of mathematics content: “there seems
to be little evidence of a clear relationship between a well-developed formal academic knowledge
of particular subjects and effective teaching in the primary phase of schooling”(p. 47). Further,
teachers could and do continue to learn in the classroom. Whether the concerns of Henderson
and Rodrigues, Hine, and Kotzee are valid is contested, since there is limited transparency of
practice within tertiary settings and in teacher preparation in particular (QAO, 2013; TEMAG,
2014). However, some insight with respect to teacher preparation orientation and practice can be
gained by examining entrance demands, delivery modes, and assessment modes.
Entry processes are similar across the nation, as is the program structure. In some institutions
an Australian Tertiary Admissions Rank (ATAR) is stipulated to enter a bachelors program;
however, universities offer alternative pathways for entry such as bonus points for
disadvantaged students, rural or local students, and Indigenous students. Other institutions
stipulate the completion of particular subjects at either high school or at a tertiary level. The most
common metric for entry into a graduate program is that four mathematics-rich tertiary courses
have been completed. Some indication of the structure of mathematics teacher preparation
processes can be gained via universities’ published course profiles. These documents outline
academic learning time and assessment formats, goals, and pre-requisite courses, and describe
each course in the program. Table 1 below illustrates the diversity of assessment pathways as
well as opportunities to learn MCK and PCK in a range of teacher preparation institutions within
Australia.
Table 1
Sample of Course Assessment Modes in Sandstone Australian Mathematics Middle Years Curriculum
Courses (The study institution is Griffith University-2016 structure)
University*
Course
code
Assessment forms
Recommended
contact face to
face
Griffith
University
EDN3024/7024
Closed book exam 60%
Classroom-based research
assignment 40%
32 hrs
University of
Queensland
EDUC6725
Review of digital resources 33%
Mathematical investigation
inquiry 33%
Resource, working with families
33%
24 hrs
Monash
University
(graduate)
EDF5017
Tasks exploring numeracy-
related issues 50%
Critical reflections on numeracy
50%
24 hrs
University of
Adelaide
EDUC4533A
Workshop activities 20%
Lesson Plan 30%
Unit Plan 40%
Attendance 10%
Prepare teaching materials 50%
4hrs/week
Queensland
University of
Technology
CRB204
Learning log 60%
Teaching plan 40%
Not stated
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University of
Sydney
EDSE3046
4000 word essay 60%
2000 word assignment 40%
32 hrs
University of
Melbourne
EDUC90457
Two reports 50% each
36 hrs
University of
Newcastle
EDUC1090
Essays/written assignments
Lesson plan
8 hrs
Key: *Sourced from online university web sites.
The condensed review of course profiles across Australia presented in Table 1 suggests that
they rarely focus on specific pedagogy or MCK in either the middle school or senior pathway.
Griffith University has a test that assesses depth of MCK and PCK as part of the assessment
package. As noted above, entry to mathematics curriculum courses is on the basis of past subjects
studied. For example, Sydney University provides an extensive list of mostly calculus subjects,
and undergraduates are expected to study between four and six of these (University of Sydney,
2016). Entry to course enrolment on the basis of past studies allows the assumption that the
trainee teachers have the necessary MCK. Acceptance of this assumption allows the mathematics
curriculum courses to focus on the development of planning and aspects of PCK via preparing
teaching materials, writing reports, reviewing digital resources, and writing essays. Some course
profiles tend to emphasise generic graduate skills and niche issues, including at the University of
Sydney where brain-based research into gender differences in adolescence is explored with
implications for practice in mathematics classrooms (University of Sydney, 2016, p. 1). Curtin
University (2016, p. 1) claims that its master’s program provides candidates with a strong
background in learning theory, curriculum development, and in providing supervision.
For assessment purposes a trainee teacher in most Australian teacher preparation institutions
is likely to write essays, and complete planning tasks such as units of work. For example, they
will be required to prepare a unit on fraction development, and in that process learn the content
and pedagogy related to teaching fractions. Similarly, a trainee teacher might prepare an
investigation of developing geometrical concepts and learn in depth the mathematics
underpinning, say, Pythagoras. It is highly unlikely that the same unit of work would cover
trigonometry. Report writing, preparing a teaching plan, writing essays, and completing a
learning log are unlikely to assess fluency across a broad range of middle school mathematics
including algebra conventions, surds, quadratics, trigonometry, basic fractions, percentages,
index notation, and whole-number computation; nor do they explain how this material might be
coherently taught without access to the internet. Thus, while essays, planning tasks, and critical
reflections are useful assessment tools, by their very nature any development of MCK or PCK
within the mathematics curriculum course will likely be narrowly focused because there is
limited time to engage with the trainee teachers in order to explore niche domains, general
pedagogical principles, as well as the detail of middle years’ content. The validity of assuming
that prior mathematics course completion is a reliable measure of depth of mathematical fluency
and understanding has been questioned previously (e.g., Ball, 1990; Ball et al., 2005; Burghes &
Geach, 2011; Hill et al., 2005; Qian & Youngs, 2016; Speer et al., 2015) but has had limited empirical
examination in Australian contexts.
With respect to the face-to-face time allocated to learning to teach mathematics, the usual
practiced in Australia is a blend of lectures and workshops. Courses such as those run by the
University of Newcastle rely on delivery via digital media.
Given this background, the aims of this study are to:
1. Document the content knowledge of middle school trainee teachers in an Australia and
a Chinese teacher education program.
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2. Document entry pathways and opportunities to learn to teach mathematics in the two
institutions.
These data are used to reflect upon teacher preparation pathways and practices in each study
institution.
Method
The methodology is a multiple instrument case study where several cases provide insight into
the issue (Cresswell, 2015; Gay, Mills, & Airasian, 2006). Holosko and Thyer (2011) have
suggested that such case study methodology is useful in investigating a number of variables and
the relationship between these in influencing behaviour. Consistent with the recommendations
of Yin (2009), several data sources including quantitative and qualitative forms are used to help
the readers appreciate the phenomena.
The data sources include documentation in the form of summaries of assessment protocols,
time delivery, and program structures in Chinese and Australian teacher preparation programs.
This detail enables the reader to compare the processes. Quantitative data describing Chinese and
Australian trainee teachers’ mathematics knowledge illustrate the different starting knowledge
of the cohorts. The statistical differences (or otherwise) between the samples are supported by
analysis of the concepts of mathematics with which trainee teachers struggled. This detail adds
relevance to the descriptive statistics in terms of the stage of preparedness for teaching children
similar concepts and processes. Supporting cultural commentary provides background content
to help explain and analyse the data. SPSS was used to calculate the means and standard
deviations of different cohorts. In terms of the differences between Chinese and Australian
overall means, statistical tests are not needed since the differences are so profound. Further, what
is important is not if the means are statistically significant, but rather, if they are educationally
significant.
Participants
The subjects were trainee middle school mathematics teachers in an Australian and a Chinese
university. The Australian university is situated in the capital of Queensland, Brisbane, and is
ranked 16 out of 29 nationally and 382 on the world ranking system (4 International Colleges &
Universities, 2015). The program and courses studied by these Australian students were
accredited by the Queensland College of Teachers (QCT), a formal statutory body that accredits
teacher training programs and registers teachers to teach. The Chinese university was located in
Wenzhou, a middle-sized coastal city with a population of 8 million. It ranked 179 out of 741
Chinese universities and 1,254 on the world ranking system (4 International Colleges &
Universities, 2015).
The sample of Chinese teacher education students was almost the entire cohort of a first-year
mathematics curriculum course (n=94) studying a 4-year bachelor degree specialising in
mathematics teaching at a normal university (“normal” originally applied to institutions that
focused on preparing school teachers and has been retained subsequent to the offering of courses
beyond education-related study). In China, mathematics teachers do not have a second subject:
they specialise in teaching mathematics. Entry to the program includes mandatory high scores in
closed-book examinations of mathematics. There was only one pathway into teacher training at
this university and all the trainee teachers were expected to teach both middle and senior
mathematics. Some of these trainee teachers would find themselves teaching primary
mathematics. A portion of the cohort was likely to choose to undertake additional study in the
form of a Master Degree in Education since schools were increasingly showing a preference to
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recruit those with such qualifications. The data were collected during a mathematics pedagogy
lecture in early 2015 and there is no reason to assume they are not indicative of the general
standard of mathematics proficiency among initial teacher education students enrolled in
mathematics curriculum courses in that institution. Several authors cited in the literature review
have noted that there is considerable uniformity with respect to teacher preparation across China
as a result of highly interventionist government policy.
In Queensland, as in the rest of Australia, there are two pathways to middle school and
subsequently senior high school teaching. The first is the undergraduate pathway. Here, students
complete specialist mathematics courses as part of an undergraduate degree, and then
curriculum courses specifically related to teacher education. As part of the curriculum courses
there are two mathematics-orientated curriculum courses: one for middle school and one for
senior mathematics. All of the undergraduate pathways prepare students to teach senior
mathematics and that includes considerable calculus associated with mathematical methods and
specialist mathematics. Unlike in China, all Queensland teachers need to have two teaching
specialities, mathematics and one other.
The second pathway for mathematics teacher registration in Australia is the graduate
diploma pathway. The intake consists of individuals who have a degree that is considered to
contain sufficient mathematics to give them the background to teach mathematics to Year 10.
About half of these trainee teachers go on to teach senior mathematics, while the remainder are
qualified to teach to Year 10; content at this level includes surds and quadratics.
The Australian subjects in this study were all enrolled in a middle school mathematics
curriculum course. They include virtually the entire enrolments of middle school trainee teachers
across two campuses in 2016 and almost all the enrolments from one campus in 2014 and 2015.
The total number of Australian trainee teachers sampled was 192. This sampling gives us
confidence that the data are representative of the case study institution.
Instruments and Data Collection
The programs and course structures of each institution were summarised in tables. These were
accessed via the published material online from each institution.
The trainee teachers’ mathematical competency with basic facts and processes was assessed
using a modified Burghes (2007) International comparative study in mathematics teacher training:
Trainee teacher primary mathematics audit part B (15 questions). This instrument is also Part A of the
Trainee Teacher Secondary Mathematics Audit (Burghes, 2007). This use gives readers the
opportunity to cross reference with the results of Burghes and Geach (2011) International
comparative study in Mathematics teacher training. Of these questions Burghes and Geach (2011, p.
7) commented:
These are the responses to the relatively straightforward questions on concepts that were also taken
by the primary participants. We would expect the secondary trainees to do well on this part of the
audit…. The audits undoubtedly stress procedural rather than conceptual mathematics… we have
gone for consistency and reliability rather than complexity.
The justification for assessing trainee teachers’ MCK via fluency and depth of content
knowledge has been outlined in the literature review. The Chinese trainee teachers were
permitted 30 minutes and the Australian trainee teachers were permitted 1 hour. The reason for
this discrepancy was that the Chinese author considered most of the questions relatively trivial
given the level of mathematics previously studied and the expectation that teachers of
mathematics would be very fluent in basic procedures and problem solving. Calculators were not
permitted and the test was closed book and supervised by one of the authors. The Australian
participants filled out additional background and attitudinal data that were not reported in this
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paper, an activity estimated to take about 10 minutes. The authors of this study did not use Part
B of the Burghes (2007) audit as the first author considered it sufficient to test to the concepts
associated with Secondary audit Part A, in part because about half the Australian sample would
only be registered to teach to Year 10 mathematics and not senior mathematics.
Results
Opportunity to Learn to Teach Mathematics; Chinese Trainee Teachers
Table 2 is a summary of courses as translated from “Cultivation plan” for the Bachelor of Science
in Mathematics and Applied Mathematics (teaching degree in the first tier university).
Table 2
Mathematics Learning Opportunity for Chinese Secondary Trainee Teachers of Mathematics
Compulsory mathematics foundation
subjects
Mathematics electives (240 hrs to be selected)
Advanced algebra 1 (64 hrs)
Advanced geometry (48 hrs)
Analytic geometry (32 hrs)
Ordinary differential equations (48 hrs)
Mathematical analysis (96 hrs)
Mathematical modelling and experiments 80 hrs)
Probability theory (48 hrs)
Abstract algebra (32 hrs)
C programming (80 hrs)
Abstract algebra (32 hrs)
Advanced algebra 2 (80 hrs)
Functions of complex variables (48 hrs)
Mathematical analysis 2 (96 hrs)
Calculation methods (64 hrs)
Mathematical analysis 3 (64 hrs)
Discrete mathematics (32 hrs)
Total (569 hrs)
Mathematical statistics (32 hrs)
Elementary number theory (32 hrs)
Function of complex variables (32 hrs)
Operations research (48 hrs)
Further studies in algebra (32 hrs)
Calculation methods (32 hrs)
Abstract algebra (32 hrs)
Further studies in mathematical analysis (32 hrs)
Function of a real variable (48 hrs)
Further studies in mathematical analysis (32 hrs)
Discrete mathematics (48 hrs)
Differential geometry (32 hrs)
Compulsory subjects make up 70% of the course requirements. In addition to the list above
there were compulsory subjects related to specific mathematics elements including pedagogy of
mathematics (48 hours), analysis of mathematics curriculum standards and text books (16 hours),
training of mathematical teaching skills (16 hours), and studies of trends of mathematics teaching
(16 hours). Chinese trainee teachers at this institution received well in excess of 600 hours of
explicit instruction related to mathematics and how to teach it. These data complement the
Burghes and Geach (2011) description of a 4-year Bachelor Diploma Degree and Certificate of
Teacher Training which included components in pedagogy, psychology, educational technology,
mathematics and applied mathematics, computer science, teaching practice, and a dissertation.
Middle school mathematics teacher preparation Norton & Zhang
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The teaching practice component included observed lessons, marking homework, and teaching
under the observation of supervising school teachers and university academics. Of relevance to
the study is that enrolment is dependent on an entrance examination including Chinese, English,
Mathematics, and Physics.
The program structure of Australian trainee teachers follows two pathways as described
above. The specific courses are listed below.
Table 3
Summary of Content Prerequisite University-based Courses for Undergraduate Pathway Students and
Entry Criteria for Graduate Pathway (4 years Bachelor Program)
Summary of content of Undergraduate Pathway to middle school and senior mathematics
teaching
Mathematics 1A
Basic differential calculus of one variable, partial derivatives, basic
vector algebra in two and three dimensions
Mathematics 1B
Probability, complex numbers, differential equations, and linear
algebra
Mathematics 2A
Multiple integrals of scalar functions, Gauss, Green, and Stokes,
Fourier series and integrals
Linear algebra
Multiples of integrals, of scalar functions, differential equations,
theorems of Gauss, Green, and Stokes
Numerical methods
and MATLAB
Maths that cannot be solved by hand including non-linear equations,
linear systems, data fitting, integration and solutions of systems of
differential equations (use of MATLAB)
Introduction to
mathematical
modelling
Mathematical models related to derivatives, rates, integrals,
optimisation, and ordinary differential equations
Graduate pathway summary of entry requirements for middle school and senior mathematics
course enrolment
Middle school
teaching only
Degree that contains at least four university-based subjects rich in
mathematics
Middle and senior
teaching
Degree that contains at least six university-based subjects rich in
mathematics
With respect to the undergraduate content courses listed above, it is typical for the lectures
to be derived online via lecture capture, and for students to attend 26 hours of tutorials. A
reasonable estimate of recommended contact time is 180 to 200 hours for Bachelor of Secondary
Education students with about 50 to 60 mathematics curriculum contact hours.
Typical of other Australian institutions, with respect to the graduate entry, the program
convenor looks for a spread of mathematics in calculus, statistics, and modelling. Thus, some of
the graduate students have majors in finance or accounting, while others are scientists, engineers,
or draftspersons. A few have doctorates in research science and even in mathematics. All of the
above Australian trainee mathematics teachers will do one 10-credit point course in middle
school mathematics curriculum. Those who wish to teach senior mathematics will undertake one
further mathematics curriculum subject. At the study site this ranges from 28 hours of contact to
24 hours depending on the lecturing academic. In 2019 it is proposed to offer 10 hours of
workshops and 15 hours of online lectures over 5 weeks for each secondary school mathematics
Middle school mathematics teacher preparation Norton & Zhang
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curriculum course. There is no entrance examination for enrolment in Australian teacher
education programs (at the graduate level) that might test the level of mathematics associated
with upper middle years’ mathematics. Depth of knowledge of mathematics is inferred from the
completion of prior courses.
Results on Basic Knowledge of Content
Summary of results from Burghes’s (2007) 20-mark (MCK) scale with standard deviation on
fundamental mathematics is provided in Table 4.
Table 4
Results on Burghes’s (2007) Test for Primary and Secondary Samples, Means and Standard Deviations
Sample
Mean/20 (Sd.)
China 2015 (n=94)
18.4 (1.5)
Australia 2014 (n=44)
8.3 (3.9)
Australia 2015 (n=30)
9.5 (4.2)
Australia 2016 (Undergraduates n=21)
12.8 (3.2)
Australia 2016 (Graduate entry n=97)
7.5 (3.8)
By way of comparison, Burghes and Geach (2011) reported their Chinese sample with a mean
of 16.6 (SD 1.6). The Australian scores in this study are less than those reported by Burghes and
Geach for England (14.1/3.5), Japan (16.3/1.8), Russia (17.3/2.0), Ukraine (15.5/2.6), and
Singapore (15.3/2.3).
We can see in Table 4 that in general the undergraduate cohort achieved higher grades. It is
hardly surprising that the undergraduates outperformed the graduate entry cohorts, since the
undergraduates had recently completed six tertiary courses focused on mathematics while
portions of the gradate pathway had not studied mathematics for extended time frames. Since
all these prospective teachers will be qualified to teach middle school mathematics and the
graduate pathway outnumber the undergraduate pathway in the order of 6:1, the overall average
Australian success rate is reported henceforth. There is no virtue in conducting statistical
significance tests on the Australian and Chinese data (clearly the significance is about p=.000);
the real question is, what is the educational significance? It is worth exploring a few of the
questions in detail to gain an appreciation of the mathematics involved.
Question 3. Let a =2, b = -1. Calculate the value of H when
=
+
.
This problem is pure algebraic procedure and involves substitution, fractions, and integers.
Fluency while working with algebraic fractions is a Year 10 expectation (ACARA, 2012). The
Chinese sample had a 94% success rate and the Australian samples averaged 45% success.
Question 5: A ball is dropped from a height of 12 metres. It bounces on the ground and
reaches
of its height. It continues to bounce this way, each time rising to
of the previous height.
What height does the ball reach after three bounces?
The solution can be found with diagrammatic modelling.
Model the problem:
First
bounce
Second
bounce
Third
bounce
Middle school mathematics teacher preparation Norton & Zhang
MERGA 13
Thus the height after the third bounce is 12×
×
×
=
or 5
.
Figure 1: Possible solution to Question 5.
A trainee teacher who found the solution by modelling such as above would be
demonstrating “mathematical knowledge for teaching” (Hill et al., 2005). A trainee teacher who
could not solve the problem via any method would be demonstrating a low level of MCK.
The fraction computation in Question 5 is Year 7 level in Australian schools (ACARA, 2012).
The Australian trainee success rate was 13% and the Chinese trainee teachers had a 90% success
rate. This question is not pure procedure; the context requires a little thought unless the student
readily recognised depreciation and index structure.
Question 6 asked the trainee teachers to factorise -7x + 12.
This question is pure procedure requiring knowledge of factorisation and integers and is
content of Year 10 (ACARA, 2012). Since the “a” value in a +bx + c is 1, students need to
consider two numbers that multiply to +12 and add to -7. Competent Year 10 students will
recognise these to be -1 and -6, and express them in factorised form (x-1)(x-6) without using a pen
to carry out any computations. The Chinese success rate was 99% and the overall Australian
success rate was 18%.
Question 7: Tom, Dick, and Harry have a sum of $575 to be shared among them. They agree
to divide it so that Tom gets $19 more than Dick, and Dick gets $17 more that Harry. How much
does Tom get?
The solution is modelled in Figure 2 so the reader can get a sense of the problem solving in
algebra contexts involved.
Modelling the problem:
Harry’s money
Dick’s money
Tom’s money
Write the equation: Let H=Harry’s share: 3H+17+17+19= 575
Simplify the equation: 3H+53=575
Solve for H (subtract from 53 both sides) 3H+53-53=575 -53
Solve for H (divide both sides by 3)
=
Do the division: H=174
Therefore Tom’s share is 174+17+19=$210
Figure 2: Possible solution method for Question 7.
Problems of this structure are typically taught in Year 8 in Australia (ACARA, 2012), but the
verbal context makes it more problem solving than pure procedure. The Chinese success rate was
86%; the Australian success rate was 33%.
Harry’s share
Harry’s share
+17
Harry’s share
+17
+19
Start 12
m.
12×
12×
×
12×
×
×
Middle school mathematics teacher preparation Norton & Zhang
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In the main, the undergraduate students were considerably stronger in problems that were
purely mechanical, such as finding and applying surd rules (Question 1), finding the cube root
(Question 2), substitution (Question 3), and factorizing a quadratics (Question 6) and the margin
of superiority was less for the problem-solving activities based on relatively simple arithmetic or
basic algebra such as the questions presented in detail above. Consistent with cognitive load
theory, the data suggest that while surface or procedural knowledge is a pre-requisite for more
advanced problem solving, it alone does not necessarily guarantee higher level problem solving.
There was no post-test subsequent to the teaching of this material in China, but for the
Australian students a number of questions were tested as part of the final assessment at the end
of the course (2016). Needless to say the results were much better. The point of the paper is not
to assess the effectiveness of an intervention, but to question whether basic content needs to be
accounted for in this institution and potentially in other Australian middle school teacher
preparation programs. Thus, the entry data are the critical data, rather than exit data after an
intervention.
Discussion
With respect to enrolment selection in Australia, the use of proxy measures of mathematical
knowledge such as those based on courses completed is typical of graduate entry to secondary
mathematics teaching, and has been criticised by earlier authors (e.g., Ball, 1990; Ball et al, 2005;
Burghes & Geach, 2011; Hill et al., 2005; Masters, 2016; Qian & Youngs, 2016; Speer et al., 2015).
The data in Table 4 provide empirical evidence to support the assertions of these authors.
The sample of assessment formats across Australia, listed in Table 1, illustrates that learning
opportunities and program structures are similar across Australia in that one or two mathematics
curriculum courses are allocated to graduate entry trainee teachers and undergraduate trainee
teachers after they complete specialist mathematics subjects. The normal modes of mathematics
curriculum assessment across Australia use take-home assignments to assess and grade students;
mostly, these assignments are associated with resource construction, essays, reflection, and lesson
planning. While these forms of assessment are valuable, by their nature they tend to focus on one
or at most a few mathematical concepts or pedagogical issues and frequently the topics are self-
selected. In this way the data in Table 1 lend support to the concerns expressed by a range of
authors (e.g., Chang, 2002; Keeling & Hersh, 2012; Kotzee, 2012; Marginson, 2002, 2006; Meyers,
2012; Sahlberg, 2006) that there may be a drift away from high demand with respect to expertise
and fluency with discipline knowledge in some Western institutions. A point of difference
between the course at the study institution and other Australian mathematics curriculum courses
surveyed is that the case study subject course assesses mathematics knowledge upon entry and
at the conclusion of the course. This final 3-hour closed-book test has components of MCK and
PCK such as capacity to diagnose student thinking from student scripts and detailed teaching
sequences for over a dozen key middle years’ concepts. It would be difficult to pass such a test
without a robust knowledge of middle years’ mathematics content and specific pedagogy.
Descriptions of the Chinese trainee students’ academic learning time indicate approximately
800 hours of pure mathematics learning (hours of lectures and workshops) prior to commencing
curriculum and pedagogy courses. Specific mathematics curriculum courses amount to 100 hours
of lectures and workshops. In addition, these beginning teachers receive ongoing professional
development and are supported by experienced teachers in the workplace. The conditions of
entry and general structure of bachelor programs conform to the Chinese processes described by
Burghes and Geach (2011) and are consistent with other analyses of Chinese teacher training
research (e.g., Fan et al., 2004; Lai & Murray, 2012; Li, 2004). Entrance to teacher preparation in
Middle school mathematics teacher preparation Norton & Zhang
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China included formal examinations in mathematics to enable prospective teachers to
demonstrate the currency of their mathematics knowledge. The academic learning in the
undergraduate pathway in the Australian institution was about a quarter of the Chinese quota.
The academic learning time for teaching of specific mathematics curriculum (pedagogy) in the
Australian institution amounted to about a third of the Chinese quota of lectures and workshops.
In Australia in general, including in the study institution, much greater use is made of self-study
and online lecture capture.
The data on mathematical knowledge of the Chinese sample very closely mirror outcomes
reported by Burghes and Geach for Chinese trainee teachers (2011). That is, for most of the
Chinese cohort the questions based on the foundations of mathematical facts and processes were
trivial. These data support the commentary of earlier researchers (e.g., Dai & Cheung, 2004; Fan
et al., 2004; Li et al., 2008; Ma, 1999; Tatto et al., 2015; Wang et al., 2004) that Chinese teachers tend
to be strong on MCK, including fluency with basic processes. According to cognitive load
theorists, the ongoing testing of mathematical content encourages Chinese trainee teachers to
commit key mathematical memory into long-term memory; that is, to progress towards meeting
Sweller’s (2016) definition of an expert. Further, the course structures illustrate that throughout
the teacher education program there are considerable opportunities to link MCK to pedagogical
practices both via the curriculum courses and in school-based internships. In other words, fluency
with school mathematics concepts can be developed into mathematical knowledge for teaching.
Cognitive load theorists (e.g., Kirschner et al., 2006; Chen et al., 2016), Asian educators (e.g., Gu,
Huang, & Marton, 2004; Huang & Leung, 2004; Lai & Murray, 2012; Li 2004; Zhang et al., 2004)
and general educational theorists (e.g., Ericsson et al., 1993; Hattie, 2009) consider deliberate
practice over significant time to be necessary to develop expertise. The Chinese teacher education
programs offer such opportunities to a greater degree than do the Australian programs.
From an accountability perspective, the data illustrated in Table 1 indicate that, in Australia,
trust and flexibility of delivery and assessment are held in high regard. A range of authors (e.g.,
Ball, 2010; Ball et al., 2005; Dinham, 2015; Elton, 2000; Hill et al., 2005; Keeling & Hersh, 2012;
Marginson, 2002, 2006; Meyers, 2012) have cited issues of governance and the role of universities
in the market place. Marginson (2006) comments that universities in Australia, including top
“sandstone” universities, have “focused more on numbers and revenues than positional value
and student quality” (p. 29). Certainly, the international commentators (e.g., Ball et al., 2005;
Burghes & Geach, 2011; Dinham, 2015; Fan et al., 2004; Sahlberg, 2006, 2007, 2011a, 2011b; Tatto
et al., 2015) recommend that a system-wide, cohesive and comprehensive approach to teacher
preparation is needed.
The results based on tests of the Australian middle school trainee teachers are little better
than those for most European trainee primary teachers, and below those for Chinese, Japanese,
and Russian trainee primary teachers (Burghes & Geach, 2011). This is concerning, not least
because European primary school teachers are generalist teachers whereas the Australian
secondary teachers will have two discipline areas. The low level of fluency with basic middle
school mathematics confirms the earlier articulation by Australian authors that there might be
reason for concern with respect to the levels of preparedness of Australian secondary teachers to
enter classroom practice (Hine, 2015; Masters, 2009, 2016; QAO, 2013; TEMAG, 2014). The data
do not mean the average graduate of this particular program will enter the classroom grossly
unprepared: some of the trainee teachers of both the undergraduate and graduate pathways
scored well on the basic entry tests, and many more learnt quite a lot of MCK as well as how to
teach it over the following 7 weeks. Some trainee teachers had histories of self-directed learning,
including the completion of doctoral studies in mathematics or science.
A number of authors in various fields (e.g., Kirchner et al., 2006; Meyers, 2012; Muller, 2009;
Muller & Taylor, 1995) claim that Western education has become hostage to the view that
Middle school mathematics teacher preparation Norton & Zhang
MERGA 16
teachers, as with other learners, can be assumed to be responsible to teach themselves or peer
teach through shared learning and everyday experiences. Poulson (2001) argued that, for primary
teachers, formal explicit discipline knowledge of mathematics might not be necessary for effective
teaching of mathematics. Further, Poulson noted that much of what a teacher needs to know in
order to teach effectively is learned on the job, including while teaching particular concepts.
Indeed, this idea is implied in the widely cited descriptor “initial teacher education” rather than
the more behaviourist “teacher training”. If “pre-service teacher education students” are just
embarking on a journey of lifelong learning, the shortcomings of not knowing the sorts of
mathematics reported in this paper become less problematic.
The potentially negative interpretation of the Australian data is only valid if it is accepted
that teachers ought to graduate with high levels of MCK, and that fluency with basic facts and
processes is understood to be essential for effective teaching. Most educational theorists and
report writers accept that this is a given (e.g., Australian Academy of Science, 2015; Ball, 1988;
Ball et al., 2005; Burghes & Geach, 2011; Goulding, Rowland, & Barber, 2002; Hill et al., 2005;
Krainer et al., 2015; Masters, 2009, 2016; TEMAG, 2014; U.S. Department of Education, 2008).
Cognitive load theorists (e.g., Chen et al., 2016; Kirschner et al., 2006; Kirschner et al., 2017;
Sweller, 2016) provide an explanation based on cognitive architecture as to why the MCK
including fluency with middle school mathematics ought to reside in a teacher’s long-term
memory. They and others (e.g., Ball et al., 2005; Englemann, 2007; Hattie, 2009) argue that,
without such knowledge, beginning teachers will be challenged to scaffold student learning,
especially for those children most in need of explicit scaffolding, the most novice of high school
students.
Conclusion
The second author was not surprised by the Chinese fluency in basic facts and processes. He was
not familiar with the terminology of cognitive load theory but instantly recognised its
manifestation in the Chinese educational process. The priority for him was to ensure that fluency
in basic facts and processes was associated with deeper problem solving and was connected to
domain-specific pedagogy. In this regard, MCK of the form tested can be regarded as what Hattie
and Donoghue (2016) described as lower level learning, to be connected via the teacher education
program to deeper mathematical understanding and creative thought associated with teaching.
This is a significant task, but there was considerable opportunity to pursue these goals.
Across Australia there is a commonality of program structure for middle school mathematics
teacher preparation in terms of the number of courses and the considerable flexibility of entry
requirements. At the course level, there is considerable flexibility with regard to delivery format,
allocated academic learning time, and assessment modes. This flexibility allows individual
academics to treat mathematics curriculum courses as capstone courses that build upon the MCK
that trainee teachers enter with. The Australian mathematics curriculum course modes of
assessment indicate that MCK related to the specific mathematics to be taught is largely assumed.
The data presented in this paper call into question the wisdom of this assumption. Testing data
at entry to the mathematics curriculum course, especially with regard to the graduate entry
pathway, indicated that most were not fluent in the mathematics they were being qualified to
teach; according to most mathematic educational theorists and cognitive load theorists, to adjust
program and course structure to account for this knowledge would seem necessary.
The challenge for the Australian academic was to attempt to build fluency and depth of
understanding in content and connect this with effective domain-specific pedagogy within the
constraints of a single mathematics curriculum course. The Australian program structures and
Middle school mathematics teacher preparation Norton & Zhang
MERGA 17
reported condensed course delivery opportunities forced the academic to prioritise what
knowledge forms are most important. In prioritizing, the authors consider the advice of cognitive
load theorists that, in the absence of substantive discipline knowledge, general pedagogy is likely
to be an empty vessel. Other English speaking Western teacher preparation nations might
consider whether similar data are prevalent across their teacher preparation institutions.
References
4 International Colleges & Universities. (2015). Universities in China by 2015 University Web
Ranking. Retrieved from https://www.4icu.org
Australian Academy of Science. (2015). Desktop review of mathematics education pedagogical
approaches and learning resources. Retrieved from www.science.org.au
Australian Curriculum, Assessment and Reporting Authority (ACARA). (2012). The Australian
Curriculum: Mathematics. Retrieved from https://www.australiancurriculum.edu.au/f-10-
curriculum/mathematics/
Australian Institute for Teaching and School Leadership (AITSL). (2012). Australian Professional
Standards for Teachers (Extract) Graduate Career Stage. Retrieved from
http://www.qct.edu.au/PDF/PSU/APST_GraduateStage.pdf
Ball, D. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8, 40-48.
Ball, D. (1990). The mathematical understandings that prospective teachers bring to teacher
education. Elementary School Journal, 90, 449-466.
Ball, D., Hill, H., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics
well enough to teach third grade, and how can we decide? American Educator, Fall. Retrieved
from www.aft.org/pubs-reports/americian_educator/fall05
Ball, S. (2010). The teacher’s soul and terrors of performativity. Journal of Educational Policy, 8(2),
215-228.
Bernstein, B. (1999). Vertical and horizontal discourse: An essay. British Journal of Sociology of
Education, 20(2), 157-173.
Bernstein, B. (2000). Pedagogy, symbolic control and identity (Rev. ed.). Lanham, Boulder, New York,
Oxford: Rowan & Littlefield Publishers, Inc.
Brown, T., McNamara, D., Hanley, V., & Jones, L. (1999). Primary student teachers’
understanding of mathematics and its teaching. British Educational Research Journal, 25(3),
299-322.
Burghes, D. (2007). International comparative study in Mathematics training. CfBT Education Trust.
Retrieved from http://www.cimt.plymouth.ac.uk/papers/icsmtt.pdf
Burghes, D., & Geach, R. (2011). International comparative study in Mathematics training:
Recommendations for initial teacher training in England. CfBT Education Trust. Retrieved from
https://www.educationdevelopmenttrust.com/~/media/EDT/Reports/Research/2011/r
-international-comparative-study-in-mathematics-teacher-training-research-2011.pdf
Cai, J. Mok, I., Reddy, V., & Stacey, K. (2016). International comparative studies in mathematics:
Lessons for improving students’ learning. Springer International Publishing. Springeropen.
Chang, H-J. (2002). Breaking the mould: An institutionalist political economy alternative to the
neo-liberal theory of the market and the state. Cambridge Journal of Economics, 26, 539-559.
Chen, O., Kalyuga, S., & Sweller, J. (2016). Relations between the worked example and generation
effects on immediate and delayed tests. Learning and Instruction, 45, 20-30.
Cresswell, J. (2015). Educational research: Planning, conducting, and evaluating quantitative and
qualitative research. Pearson.
Middle school mathematics teacher preparation Norton & Zhang
MERGA 18
Curtin University. (2016). Master of science and mathematics education: (Post graduate). Retrieved
from http://courses.curtin.edu.au/course_overview/postgraduate/Master-Science-
Mathematics-Education
Dai, Q., & Cheung, K. (2004). The wisdom of traditional Mathematics teaching in China. In L. Fan,
N-Y Wong, J. Cai, & S. Li (Eds.), How Chinese learn Mathematics: Perspectives from insiders (pp.
3-42). New Jersey: World Scientific.
Dinham, S. (2015). The worst of both worlds: How U.S. and U.K. models are influencing
Australian education. Education Policy Analysis, Archives, 23(49). Retrieved from
http://epaa.asu.edu/ojs/
Elton, L. (2000). The UK research assessment exercise: Unintended consequences. Higher
Education Quarterly, 54(3), 274-283.
Englemann, S. (2007). Teaching needy kids in our backward system. Eugene: ADI Press.
Ericsson, K., Krampe, R., & Tesch-Romer, C. (1993). The role of deliberate practice in the
acquisition of expert performance. Psychological Review, 100(3), 363-406.
Fan, L., Miao, Z., & Mok, A. (2004). How Chinese teachers teach Mathematics and pursue
professional development: Perspectives from contemporary international research. In L.
Fan, N-Y Wong, J. Cai, & S. Li (Eds.), How Chinese learn Mathematics: Perspectives from insiders
(pp. 43-70). New Jersey: World Scientific.
Gay, L., Mills, G., & Airasian, P. (2006). Educational research: Competencies for analysis and
application. Upper Saddle River: Pearson, Merrill Prentice Hall.
Gess-Newsome, J. (2013). Pedagogical content knowledge. In J. Hattie & E. Anderman (Eds.),
International guide to student achievement (pp. 257-259). NY: Routledge.
Goulding, M., Rowland, T., & Barber, P. (2002). Does it matter? Primary teacher trainees’ subject
knowledge in mathematics. British Educational Research Journal, 28(5), 689-704.
Gregg, J. (1995). The tensions and contradictions of the school mathematics tradition. Journal for
Research in Mathematics Education, 26(5), 442-466.
Griffith University. (2016). Course list. Retrieved from
https://degrees.griffith.edu.au/Program/1051/Courses/Domestic#0000008794
Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting
effective mathematics learning. In L. Fan, N-Y Wong, J. Cai, & S. Li (Eds.), How Chinese learn
Mathematics: Perspectives from insiders (pp. 309-347). New Jersey: World Scientific.
Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. London:
Routledge.
Hattie, J., & Donoghue, G. (2016). Learning strategies: A synthesis and conceptual model. npj
Science of Learning, 1. Retrieved from http://www.nature.com/npjcilearn
Henderson, S., & Rodrigues, S. (2008). Scottish student primary teachers’ levels of mathematical
competence and confidence for teaching mathematics: Some implications for national
qualifications and initial teacher education. Journal of Education for Teaching, 34(2), 93-107.
Henson, R. K. (2001). Teacher self-efficacy: Substantive implications and measurement dilemmas. Paper
presented at the Annual Meeting of the Educational Research Exchange, College Station, TX.
Hill, H., Rowan, B., & Ball, D. (2005). Effects of teacher’s mathematical knowledge for teaching
on student achievement. American Educational Research Journal, 42(2), 371-406.
Hine, G. (2015). Strengthening pre-service teachers’ mathematical content knowledge. Journal of
University Teaching and Learning Practice, 12(4). Retrieved from
http://researchonline.nd.edu.au/cgi/viewcontent.cgi?article=1159&context=edu_article
Holosko, M., & Thyer, B. (2011). Pocket glossary for commonly used research terms. Thousand Oaks,
CA: Sage.
Huang, R., & Leung, K. (2004). Cracking the paradox of Chinese learners: Looking into
mathematics classrooms in Hong Kong and Shanghai. In L. Fan, N-Y Wong, J. Cai, & S. Li
Middle school mathematics teacher preparation Norton & Zhang
MERGA 19
(Eds.), How Chinese learn Mathematics: Perspectives from insiders (pp. 348-381). New Jersey:
World Scientific.
Keeling, R., & Hersh, R. (2012). We’re losing our minds. Rethinking American higher education. New
York: Palgrave Macmillan.
Kirschner, P., Sweller, J., & Clark, R. (2006). Why minimal guidance during instruction does not
work: An analysis of the failure of constructivist, discovery, problem-based, experiential,
and inquiry-based teaching. Educational Psychologist, 41(2), 75-86.
Kirschner, P., Vershgaffel, L., Star, J., & Van Dooren, W. (2017). There is more variation within
than across domains: An interview with Paul A. Kirschner about applying cognitive
psychology-based instructional design principles in mathematics teaching and learning.
Mathematics Education, 49, 637-643.
Klein, D. (2005). The state of state Math standards. Thomas B. Fordham Foundation.
Kotzee, B. (2012). Expertise, fluency and social realism about professional knowledge. Journal of
Education and Work. Retrieved from
http://www.tandfonline.com/doi/abs/10.1080/13639080.2012.738291
Krainer, K., Hsieh, F., Peck, R., & Tatto, M. (2015). The TEDS-M: Important issues, results and
questions. In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical
Education (pp. 99-121). Retrieved from http://link.springer.com/chapter/10.1007%2F978-3-
319-12688-3_10#page-1,
Krauss, S., Brunner, M., Kunter, M., Baumert, J., Blum, W., Neurrand, M., & Jordan, A. (2008).
Pedagogical content knowledge and content knowledge of secondary Mathematics teachers.
Journal of Educational Psychology, 100(3), 716-725.
Lai, M., & Murray, S. (2012). Teaching with procedural variation: A Chinese way of promoting
deep understanding of mathematics. International Journal of Mathematics Teaching and Learning.
Retrieved from http://www.cimt.plymouth.ac.uk/journal/
Lannin, J., Webb, M., Chval, K., Arbaugh, F., Hicks, S., Taylor, C., & Burton, R. (2013). The
development of beginning mathematics teacher pedagogical content knowledge. Journal of
Mathematics Teacher Education, 16, 403-426.
Li, J. (2004). Chinese cultural model of learning. In L. Fan, N-Y Wong, J. Cai, & S. Li (Eds.), How
Chinese learn Mathematics: Perspectives from insiders (pp. 124-156). New Jersey: World
Scientific.
Li, Y., Zhao, D., Huang, R., & Ma, Y. (2008). Mathematics preparation of elementary teachers in
China: Changes and issues. Journal of Mathematics Teacher Education, 11, 417-430.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental
mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates, Inc.
Marginson, S. (2002). Nation-building universities in a global environment: The case of Australia.
Higher Education, 43, 409-428.
Marginson, S. (2006). Dynamics of national and global competition in higher education. Higher
Education, 52, 1-39.
Masters, G. (2009). A shared challenge: Improving literacy, numeracy and science learning in Queensland
primary schools. Australian Council for Educational Research. Retrieved from
http://education.qld.gov.au/mastersreview/
Masters, G. (2016). Policy insights: Five challenges in Australian school education. Retrieved from
https://research.acer.edu.au/cgi/viewcontent.cgi?article=1004&context=policyinsights
McEwan, H., Bull, B. (1991). The pedagogic nature of subject matter knowledge. American
Educational Research Journal, 28, 316-334.
Meyers, D. (2012). Australian universities: A portrait of decline. Retrieved from
http://www.australian universities.id.au/
Middle school mathematics teacher preparation Norton & Zhang
MERGA 20
Monash University. (2016). EDF 1206-Mathematics education 1. Retrieved from
http://www.monash.edu.au/pubs/2017handbooks/units/EDF5017
Muller, J. (2000). Reclaiming knowledge: Social theory, curriculum and education policy. London and
New York: Routledge Falmer.
Muller, J. (2009). Forms of knowledge and curriculum coherence. Journal of Education and Work.
Retrieved from http://www.tandfonline.com/loi/cjew20
Muller, J., & Taylor, N. (1995). Schooling and everyday life: Knowledges sacred and profane.
Social Epistemology: A Journal of Knowledge, Culture and Policy, 9(3), 257-275. Retrieved from
http://dx.doi.org/10.1080/02691729508578791
Organization for Economic Co-operation and Development (OECD). (2014). Education at a glance,
2014. Retrieved from http://www.oecd.org/edu/Education-at-a-Glance-2014.pdf/
Owen, E., & Sweller, J. (1989). Should problem solving be used as a learning device in
mathematics? Journal for Research in Mathematics Education, 20(3), 322-328.
Poulson, L. (2001). Paradigm lost? Subject knowledge, primary teachers and education policy.
British Journal of Educational Studies, 49(1), 40-55.
Qian, H., & Youngs, P. (2016). The effect of teacher education programs on future elementary
mathematics teachers’ knowledge: A five country analysis using TEDS-M data. Journal of
Mathematics Teacher Education, 19, 371-396.
Queensland Audit Office. (2013). Supply of specialist subject teachers in secondary schools. Report to
Parliament 2: 2013-2014. Retrieved from www.qao.qld.gov.au
Queensland University of Technology. (2016a). Bachelor of Education Secondary- Mathematics.
Retrieved from https://www.qut.edu.au/study/courses/bachelor-of-education-
secondary/bachelor-of-education-secondary-mathematics
Queensland University of Technology (2016b). Mathematics Curriculum Studies 1. Retrieved from
https://www.qut.edu.au/study/courses/bachelor-of-education-secondary/bachelor-of-
education-secondary-mathematics
Sahlberg, P. (2006). Education reform for raising economic competitiveness. Journal of Educational
Change. Retrieved from http://link.springer.com/article/10.1007/s10833-005-4884-6
Sahlberg, P. (2007). Education policies for raising student learning: The Finnish approach. Journal
of Education Policy, 22(2), 147-171.
Sahlberg, P. (2011a). Lessons from Finland. The Professional Educator, 34-38. Retrieved from
http://pasisahlberg.com/wp-content/uploads/2013/01/Lessons-from-Finland-AE-2011.pdf
Sahlberg, P. (2011b). PISA in Finland: An education miracle or an obstacle to change? Centre for
Education Policy Studies Journal, 1(3), 119-140.
Sahlberg, P., & Oldroyd, D. (2010). Pedagogy for economic competitiveness and sustainable
development. European Journal of Education, 45(2)280-299.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational
Researcher, 15(2), 4-14.
Speer, N., King, K., & Howell, H. (2015). Definitions of mathematical knowledge for teaching:
Using these constructs in research on secondary and college mathematics teachers. Journal of
Mathematics Teacher Education, 18, 105-122.
Stones, E. (1992). Quality teaching: A sample of cases. London and New York: Routledge.
Sweller, J. (2016). Working memory, long-term memory, and instructional design. Journal of
Applied Research in Memory and Cognition, 5, 360-367.
Tatto, M., Rodriguez, M., & Lu, Y. (2015). The influence of teacher education on mathematics
teaching knowledge: Local implementation of global ideas. International Perspectives on
Education and Society, 27, 279-331. Retrieved from
http://www.emeraldinsight.com/doi/pdfplus/10.1108/S1479-367920140000027004/
Middle school mathematics teacher preparation Norton & Zhang
MERGA 21
Tatto, M., Schwille, J., Senk, S., Ingvarason, L., Peck, R., & Rowley, G. (2008). Teacher education and
development study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and
secondary mathematics. Conceptual framework. East Lansing, MI: Teacher Education and
Development International Study Center, College of Education, Michigan University.
Teacher Education Ministerial Advisory Group. (2014). Action now! Classroom ready teachers.
Retrieved from
http://docs.education.gov.au/system/files/doc/other/action_now_classroom_ready_tea
chers_print.pdf
Tricot, A., & Sweller, J. (2014). Domain-specific knowlede and why teaching generic skills does
not work. Educational Psychology Review, 26, 265-283.
University of Adelaide (2017). Senior Mathematics Curriculum and Methodology A. Retrieved from
http://www.adelaide.edu.au/course-outlines/105784/1/quad-1/2017
University of Melbourne. (2016). Learning area mathematics 1. Retrieved from
https://handbook.unimelb.edu.au/view/2017/EDUC90457
University of Newcastle. (2016). Specialist Studies in Mathematics 1. Retrieved from
https://www.newcastle.edu.au/course/EDUC1090
University of Queensland. (2006). Mathematics curriculum foundations. Retrieved from
https://www.uq.edu.au/study/course.html?course_code=EDUC6725
University of Sydney. (2016). Teaching Mathematics 1B. Retrieved from
http://sydney.edu.au/courses/uos/EDSE3046
US Department of Education. (2008). Success: The final report of the National Mathematics Advisory
Panel. Retrieved from
http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf/
Wang, T., Cai, J., & Hwang, S. (2004). Achieving coherence in the mathematics classroom:
Towards a framework for examining instructional coherence. In L. Fan, N-Y Wong, J. Cai, &
S. Li (Eds.), How Chinese learn Mathematics: Perspectives from insiders (pp. 111- 148). New
Jersey: World Scientific.
Watt, H., & Richardson, P. (2013). Teacher motivation and student achievement outcomes. In J.
Hattie & E. Anderman (Eds.), International guide to student achievement (pp. 271-273). New
York: Routledge.
Wragg, E. C., Bennett, S. N., & Carre, C. (1989). Primary teachers and the national curriculum.
Research Papers in Education, 4(1), 17-45.
Yin, R. (2009). Case study research: Design and methods. Thousand Oaks, CA: Sage.
Zhang, Q., & Stephens, M. (2013). Utilising a construct of teacher capacity to examine national
curriculum reform in mathematics. Mathematics Education Research Journal, 25, 481-502.
Zhang, D., Li, S., & Tang, R. (2004). The “two basics”: Mathematics teaching and learning in
mainland China. In L. Fan, N-Y Wong, J. Cai, & S. Li (Eds.), How Chinese learn Mathematics:
Perspectives from insiders (pp. 189-207). New Jersey: World Scientific.
Authors
Stephen Norton
Griffith University
Australia
s.norton@griffith.edu.au
Qinqiong Zhang
Wenzhou University
People’s Republic of China
qqzhang922@126.com
